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Research article

A critical point theorem for a class of non-differentiable functionals with applications

  • Received: 10 January 2020 Accepted: 12 May 2020 Published: 19 May 2020
  • MSC : 35B38, 49J52

  • This paper presents a multiplicity theorem for a kind of non-smooth functionals. The proof of this theorem relies on a suitable deformation lemma and the perturbation methods. We also apply this result to prove a multiplicity theorem for elliptic variational-hemivariational inequality problems.

    Citation: Yan Ning, Daowei Lu. A critical point theorem for a class of non-differentiable functionals with applications[J]. AIMS Mathematics, 2020, 5(5): 4466-4481. doi: 10.3934/math.2020287

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  • This paper presents a multiplicity theorem for a kind of non-smooth functionals. The proof of this theorem relies on a suitable deformation lemma and the perturbation methods. We also apply this result to prove a multiplicity theorem for elliptic variational-hemivariational inequality problems.


    The mountain pass theorem of Ambrosetti and Rabinowitz for C1 functions plays an essential role in the area of nonlinear analysis. We are interested in the monograph [22] of Motreanu and Panagiotopoulos in which they established a new version of the mounbtain pass theorem [22,Theorem 3.2] for the functionals f from Banach space X to R{+} satisfying the following hypothesis:

    (Hf):f(x)=Φ(x)+Ψ(x) for all xX, where Φ:XR is locally Lipschitz continuous while Ψ:XR{+} is convex, proper, and lower semi-continuous.

    We call xX is a critical point of f if x solves the following problem:

    Φ0(x;zx)+Ψ(z)Ψ(x)0,     zX, (1.1)

    where Φ0(x;zx) is the generalized directional derivative of Φ at x in the direction zx (see [6] for detail).

    Recall that f satisfies the (PS)c condition if any sequence {xn}X for which limn+f(xn)=c and

    Φ0(xn;zxn)+Ψ(z)Ψ(xn)ϵnzxn,  nN,zX

    where ϵn0+, possesses a convergent subsequence.

    When (PS)c holds true at any level c we simply write (PS)f in place of (PS)c.

    Inequality (1.1) is usually called variational-hemivariational inequality, which has been exploited for mathematically formulating several engineering, besides mechanical questions and extensively studied from many points of view in the latest years [1,22,23].

    Variational-hemivariational inequalities can be studied in the framework of a general critical point theory which combines features of the classical convex analysis and of the theory of generalized gradients for locally Lipschitz functions. Such inequalities represent a very general pattern for several kinds of variational problems. Indeed, if ΦC1(X,R), the problem (1.1) is reduced to a variational inequality and the relevant critical point theory as well as significant applications are developed in [25]; if Ψ0, then (1.1) coincides with the problem treated by Chang in [5] which is called differential equations with discontinuous nonlinearities; differential inclusions (see [12]) and special non-smooth problems with constraints (see [13]) can be considered as special cases of variational-hemivariational inequality. Finally, when both ΦC1(X,R) and Ψ0, the problem (1.1) becomes the Euler equation Φ(u)=0 and the theory is classical. For the new results on this topic, see the excellent overview in [4,7,10,11,14,15,17,18,19,26,27].

    Chang in [5] established the critical point theory for non-differentiable functionals and represented some applications to partial differential equations with discontinuous nonlinearities. Marano and Motreanu [20] obtained a critical points theorem which extends the variational principle of Ricceri to variational-hemivariational inequalities and semilinear elliptic eigenvalue problems with discontinuous nonlinearities. The critical point theorem in presence of splitting was established by Brˊezis-Nirenberg [3]. Subsequently Livrea, Marano and Motreanu [16] extended it to Motreanu-Panagiotopoulos' setting under the the following structural hypothesis (Hf):

    (Hf): f(x)=Φ(x)+Ψ(x) for all xX, where Φ:XR is locally Lipschitz continuous while Ψ:XR{+} is convex, proper, and lower semi-continuous, and Ψ is continuous on any nonempty compact set AX such that supxAΨ(x)<+.

    And they applied the conclusions to an elliptic variational-hemivariational inequality.

    Motivatied by the above cited papers, we try to prove a multiplicity theorem of functions f fulfilling the structural hypothesis (Hf), mountain pass geometry and the bounded from below conditions. In Section 2, we will recall some basic definitions and preliminaries. The essential tool used in the proof is a general deformation lemma, which will be set forth in Section 3. Section 4 presents our main result, a new critical point theorem.

    In the last section we consider an application to the elliptic variational-hemivariational inequality:

    (Pλ): Find uKλ such that for all vKλ,

    Ωu(x)(vu)(x)dxΩa(x)u(x)(vu)(x)dxλG0(u;vu),

    where λ>0, Kλ is convex and closed in H10(Ω), g:RR is locally bounded and measurable, and the functions G:RR and G:H10(Ω)R given by

    G(ξ)=ξ0g(t)dt,   ξR,    G(u)=ΩG(u(x))dx,    uH10(Ω),

    respectively, are well defined and locally Lipschitz continuous. Some results of [16] are improved.

    Let (X,) be a reflexive Banach space. Denote by B(x,δ):={zX:zx<δ} as well as Bδ:=B(0,δ). The symbol [x,z] denote the segment joining x to z, namely [x,z]:={(1t)x+tz:t[0,1]}, and (x,z]:=[x,z]{x}. We denote by X the dual space of X, while , stands for the duality pairing between X and X. A functional φ:XR{+} is proper if Dφ={xX:φ(x)<}. Functional Φ:XR is called locally Lipchitz continuous if for every xX there exists a neighborhood Vx of x and a constant Lx0 such that

    |Φ(z)Φ(w)|Lxzw, z,wVx.

    Let Φ0(x;z) be the generalized directional derivative of Φ at x along the direction z, i.e.,

    Φ0(x;z):=lim supwx,t0+Φ(w+tz)Φ(w)t.

    The generalized gradient of the function Φ at x, denoted by Φ(x), is the set

    Φ(x):={xX:x,zΦ0(x;z), zX}.

    The mapping zΦ0(x;z) is positively homogeneous and sub-additive, thus, due to the Hahn-Banach theorem, the set Φ(x) is nonempty. In the sequel, we state the main properties of the generalized directional derivatives and the generalized gradients:

    1) For each xX, Φ(x) is a nonempty, convex in addition to weak compact subset of X.

    2) For each x,zX,Φ0(x,z) is upper semicontinuous on X×X.

    3) For each x,zX, we have Φ0(x;z)=max{x,z; xΦ(x)}.

    4) If Φ attains a local minimum or maximum at x, then 0Φ(x).

    5) The function mΦ(x)=min{xX,xΦ(x)} exists and is lower semi-continuous.

    Let Ψ:XR{+} be convex, proper, and lower semi-continuous. Set DΨ={xX:Φ(x)<+}, then Ψ is continuous in int(DΨ) (see [8]). To simplify notation, denote by Ψ(x) the subdifferential of Ψ at the point xX in the sense of convex analysis, while DΨ={xX:Ψ(x)}. By [8], int(DΨ)=int(DΨ), Ψ(x) is convex and weak closed.

    Let f be a function on X satisfying the hypothesis(Hf), aR. Define

    Ka(f)={xX:f(x)=a, x is a critical point of f},fa={xX:f(x)a},fa={xX:f(x)a}.

    For every ϵ,r>0, we introduce the set

    Fra,ϵ={xX:xr+1,and |f(x)a|ϵ},

    it is easy to see that Fra,ϵ is closed.

    In this section we establish a deformation lemma for the functions satisfying the hypothesis (Hf).

    Lemma 3.1. Suppose xint(DΨ). Then for every xnx in X and every znΨ(xn),nN, there exists zΨ(x) as well as a subsequence {zrn} of {zn} such that zrnz in X.

    For the proof the reader could refer to [21,Remark 2.1].

    Lemma 3.2. Let f be a function satisfying (Hf). Assume that there exist constants ϵ>0,r>0 and aR such that Fra,ϵ,Fra,ϵint(DΨ), and

    inf{x+z:xΦ(x),zΨ(x),xFra,ϵ}>2ϵ.

    Then for every xFra,ϵ, there exists ξxX such that

    ξx=1,x+z,ξx>2ϵ,forall  xΦ(x), zΨ(x).  (3.1)

    Proof. Since inf{x+z:xΦ(x), zΨ(x), xFra,ϵ}>2ϵ, there exists an ϵ0>0 such that for every xFra,ϵ,xΦ(x),zΨ(x), we have x+zX2ϵ+ϵ0.

    Fix an xFra,ϵ, since Φ(x) and Ψ(x) are nonempty and convex, so is Φ(x)+Ψ(x). As X is reflexive, Φ(x) is weak compact and Ψ(x) is weak closed, then Φ(x)+Ψ(x) is closed.

    Note that 0Φ(x)+Ψ(x). By [2, Corollary 3.20], we have uΦ(x),vΨ(x) satisfying

    Bδ(Φ(x)+Ψ(x))=, where δ=u+vX>0.

    Now the Hahn-Banach theorem provides a point ξxX with ξx=1 and whenever xΦ(x),zΨ(x),

    x+z,ξxw,ξx,  wBδ.

    Since u+vX=u+vXξx=max{w,ξx,w¯Bδ}, the above inequality and Lemma 3.1 lead to

    x+z,ξxu+vX2ϵ+ϵ0>2ϵ,   xΦ(x), zΨ(x).

    The proof is completed.

    Lemma 3.3 Under the conditions of Lemma 3.2, for every xFra,ϵ, there exists a δx>0 such that

    x+z,ξx>2ϵ,   xΦ(x),zΨ(x),  x, xB(x,δx),  (3.2)

    where ξx is given by Lemma 3.2.

    Proof. If the conclusion were false, then we could find xFra,ϵ,{xn},{xn}X and {xn},{zn}X such that

    xnx,xnΦ(xn),   nN;  (3.3)
    xnx,znΨ(xn),    nN;  (3.4)
    xn+zn,ξx2ϵ,    nN.  (3.5)

    Due to the reflexivity of X and (3.3), Proposition 2.1.2 of [22] yields xX such that xnx in X, where a subsequence is considered when necessary, while Proposition 2.1.5 of [22] forces xΦ(x). Since xFra,ϵint(DΨ)=int(DΨ), combining (3.4) with Lemma 3.1, we obtain, up to subsequences, znz for some zΨ(x). Now from (3.5) it follows, as n+, x+z,ξx2ϵ. However, this contradicts (3.1).

    Theorem 3.4 Let f be a function satisfying (Hf), assume that there exist constants ϵ>0,r>0 and aR such that Fra,ϵ,Fra,ϵint(DΨ), and

    inf{x+z:xΦ(x),zΨ(x),xFra,ϵ}>2ϵ.

    Then there exists a continuous mapping η:XX with the following properties:

    (1)η:X X isahomeomorphism;(2)η(x)=x whenever |f(x)a|2ϵ;(3)η(x)x1,    xX;(4)f(η(x))f(x),     xX;(5)η(DΨ)DΨ;(6)f(η(x))aϵ,  xX provided xr and f(x)a+ϵ.

    Proof. The family of balls B={B(x,δx):xFra,ϵ} constructed through Lemma 3.3 represents an open covering of Fra,ϵ, and the assumptions ensure that Fra,ϵ is a nonempty para-compact set because it is closed. So B possesses an open locally finite refinement V={Vi;iI}. Moreover, to each iI there corresponds ξiX such that ξi=1 as well as

    x+z,ξi>2ϵ,    xΦ(x),zΨ(x),  x,xVi.  (3.6)

    Shrink V to an open locally finite covering W={Wi;iI} fulfilling for every iI, WiVi ([9,Theorems Ⅷ 2.2 and Theorems Ⅶ 6.1]) with Wi is convex, and f|Wi() is Lipschitz continuous satisfying

    a2ϵ<f(x)<a+2ϵ,     xWi.  (3.7)

    Set

    W=iIWi,di(x)=d(x,XWi),ρi(x)=di(x)jIdj(x),  xW, iI,Θ(x)={iIρi(x)ξi,  if xW,0,   otherwise,l(x)=d(x,XW)d(x,XW)+d(x,Fra,ϵ),  xX,V(x)=l(x)Θ(x),  xX,V(x)=l(x)Θ(x),  xX.

    We observe that V:XX is locally Lipschitz continuous and

    V(x)1,  xX.  (3.8)

    The existence-uniqueness theorem for ordinary differential equations provides a mapping σC0(R×X,X) such that

    dσ(t,x)dt=V(σ(t,x)),  (t,x)R×X, σ(0,x)=x.

    We claim that

    for every xX, the function tf(σ(t,x)) is non-increasing on R.  (3.9)

    In fact, if xXW, V(x)=0 and thus σ(,x) is constant and (3.9) holds true. In the case xW, we start by noting that σ(R,x)W. Indeed setting T=sup{t>0:σ((t,t),x)W}, assume by contradiction that T<+. Hence we have Wx=σ(T,x)=limtTσ(t,x)W. Then the Cauchy problem

    d˜σ(t)dt=V(˜σ(t)),  tR, ˜σ(T)=Wx

    admits the constant solution ˜σ()Wx, as does tσ(t,x) for tT, which is against the uniqueness of solutions.

    Fixing tR, we know that σ(t,x)W and due to the local finiteness of W, the set J={iI:σ(t,x)Wi} is finite. It follows that ˜W=iJWi is a convex, open neighborhood of σ(t,x), and there exists δ>0 such that

     σ((tδ,t+δ),x)˜W, and σ((tδ,t+δ),x)(iIJWi)=.  (3.10)

    For arbitrary t,t(tδ,t+δ) with t<t. Lebourg's mean value theorem provides y(σ(t,x),σ(t,x)), xΦ(y), zΨ(y) satisfying

    f(σ(t,x))f(σ(t,x))=x+z,σ(t,x)σ(t,x)=ttx+z,V(σ(τ,x))dτ<2ϵttl(σ(τ,x))jJρj(σ(τ,x))dτ=2ϵttl(σ(τ,x))dτ,

    where (3.6) and (3.10) have been used. Given p,q[t,t+1] with p<q, a standard compactness argument and the above estimate enable us to find t1,t2,...,ts[t,t+1] with p=t1<t2<...<ts=q such that

    f(σ(ti,x))f(σ(ti1,x))<2ϵtiti1l(σ(τ,x))dτ,

    for all i=1,2,...,s. It turns out that

    f(σ(q,x))f(σ(p,x))=si=1[f(σ(ti,x))f(σ(ti1,x))]<2ϵqpl(σ(τ,x))dτ<0, (3.11)

    which establish (3.9). Now we define

    η(x)=σ(1,x),  xX.

    From the general theory of ordinary differential equations it is well known that η:XX is a homeomorphism.

    Since (3.7) renders {xX:|f(x)a|2ϵ}XW when xXW, V(x)=0, thus σ(,x) is constant, and η(x)=σ(1,x)=σ(0,x)=x. We then deduce property (2).

    From (3.8), for all xX, we have

    η(x)x=σ(1,x)σ(0,x)=10V(σ(τ,x))dτ10V(σ(τ,x))dτ1,

    i.e., property (3) holds true.

    Since f(η(x))=f(σ(1,x))f(σ(0,x))=f(x), for all xX, the property (4) holds true.

    For every xDΨ, there is f(x)<+. Since (4) holds, f(η(x))f(x)<+, so η(x)DΨ. i.e. η(DΨ)DΨ, (5) holds true.

    In order to prove (6), let xX with xr and f(x)a+ϵ. If f(x)aϵ, (6) follows from (4) immediately. In case aϵ<f(x)a+ϵ, we argue by contradiction. Suppose

    aϵ<f(η(x))=f(σ(1,x))f(σ(t,x))f(x)a+ϵ,  t[0,1].  (3.12)

    In addition, through (3.8), for all t[0,1] we have

    σ(t,x)x+σ(t,x)xr+t0dσ(τ,x)dτdτr+t0V(σ(τ,x))dτr+1.

    Consequently, σ([0,1],x)Fra,ϵ, which forces l(σ(,x))|[0,1]1. Then (3.11) with p=0 and q=1 reads as

    f(η(x))f(x)<2ϵ.        (3.13)

    Combining (3.12) and (3.13) gives

    aϵ<f(η(x))<f(x)2ϵaϵ,

    which is a contradiction.

    Fix v0, v1DΨ. Consider the following set of paths

    Γ={γC0([0,1],X):γ(0)=v0, γ(1)=v1}, (4.1)

    and a function f:XR{+} which verifies hypothesis (Hf). Set

    c=infγΓsupt[0,1]f(γ(t)). (4.2)

    Theorem 4.1 Suppose f:XR{+} satisfies (Hf) and (PS)f. Assume in addition that

    (i1)f is bounded below and coercive, and put α=infxXf(x);

    (i2)α<max{f(v0),f(v1)}c, v0v1 and for every γΓ, there exists t(0,1) such that f(γ(t))max{f(v0),f(v1)};

    (i3) for every aR, there exist r>0 and ϵ0>0 such that Fra,ϵ0int(DΨ).

    Then the function f possesses at least two critical points.

    Proof. Since f is coercive, for every xX with c1f(x)c+1, there exists a constant k such that

    xk. (4.3)

    From (i3), for c and k>0, there is ϵ0>0 such that

    Fkc,ϵ0int(DΨ). (4.4)

    Without loss of generality, we assume ϵ0<1. Let

    cϵ=infγΓsupt[0,1][f(γ(t))+ϵd(t)], (4.5)

    where 0<ϵ<12ϵ0 is arbitrary, and d(t)=min{t,1t}, t[0,1].

    From (i2), we can easily verify that

    ccϵ<c+ϵ.

    Since for every γΓ, there exists t0(0,1) such that f(γ(t0))max{f(v0),f(v1)}, one has

    supt[0,1](f(γ(t))+ϵd(t))f(γ(t0))+ϵd(t0)>f(γ(t0))max{f(v0),f(v1)},

    thus cϵ>max{f(v0),f(v1)} for every ϵ(0,12ϵ0).

    We claim that for every ϵ(0,12ϵ0) satisfying that ϵ<12(cϵmax{f(v0),f(v1)}) there holds

    inf{x+z:xΦ(x), zΨ(x), xFkcϵ,ϵ}2ϵ. (4.6)

    Due to the definition of ˆϵ, for every ϵ(0,ˆϵ), we have

    cϵ0<cϵ<cϵϵf(x)cϵ+ϵ<c+2ϵ<c+ϵ0

    and xk. It is straightforward to verify that Fkcϵ,ϵint(DΨ).

    To show (4.6), we argue by contradiction. If it was not true, then we would find ϵ(0,ˆϵ) for which Theorem 3.4 can be applied with a=cϵ and r=k. So there would exist a continuous mapping η:XX with the properties (1)–(6) formulated in Theorem 3.4. By the definition of ˆϵ, there is γϵΓ such that

    cϵsupt[0,1][f(γϵ(t))+ϵd(t)]<cϵ+ϵ,

    which easy to verify that

    cϵϵ<supt[0,1]f(γϵ(t))<cϵ+ϵ,

    and use the definition of ˆϵ again, it is straightforward to verify that η(γϵ())Γ for every ϵ. Hence, in view of (4.5), we may consider a sequence {sϵ} in [0,1] such that

    cϵϵ<f(η(γϵ(sϵ))),cϵϵ<f(γϵ(sϵ))<cϵ+ϵ. (4.7)

    Since cϵ+ϵ<c+2ϵ<c+1 and cϵϵ>cϵ>c1, it implies that γϵ(sϵ)k.

    Exploiting (4.7) and property (6), we achieve the contradiction

    cϵϵ<f(η(γϵ(sϵ)))cϵϵ.

    thereby (4.6) holds true.

    By virtue of (4.6), for every xX and all nN sufficiently large, there exists xnFkc1n,1n, xnΦ(xn),znΨ(xn) such that

       xn+zn<3n,

    and

    c1n<c1n1nf(xn)c1n+1n<c+2n.

    This guarantees that

    xnk+1,c1n<f(xn)<c+2n,

    and

    Φ0(xn;xxn)+Ψ(x)Ψ(xn)xn+zn,xxnxn+znxxn>3nxxn.

    Since f satisfies the (PS)f condition, there is an ¯xX such that xn¯x in X, where a subsequence is considered when necessary. At this point, ¯x is a critical point of f, and ¯xKc(f).

    Next we prove that f possesses a global minimum point x0X. Since by (i1) and the condition (PS)f, each minimizing sequence for f possesses a convergent subsequence (see[16]), the function f must attain its minimum at some point x0X.

    Due to f(x0)=α<c=f(ˉx), x0ˉx, which completes the proof.

    Remark 4.2 By the above proof, one can find that under the conditions of Theorem 4.1, when aα, there exist r>0, ϵ>0 such that Fra,ϵ. By the coercivity of f, if Ψ is convex and continuous, the condition (i3) obviously holds.

    In this section we use Theorem 4.1 to discuss an elliptic variational-hemivariational inequality in the sense of Panagiotopoulos [24].

    Let Ω be a nonempty, bounded, open subset of RN (N3) with smooth boundary Ω. Denote by H10(Ω) the usual Sobolev space with norm

    u=(Ω|u(x)|2dx)12.

    It's well known that for p[1,2], 2=2N/(N2), there exists a positive constant Cp such that

    uLp(Ω)Cpu,   uH10(Ω).    (5.1)

    Given a function aL(Ω) satisfying a(x)0 for a.e. xΩ. Let

    β=essinfxΩa(x)0.

    If g:RR satisfies the condition

    (g1)  g is locally bounded and measurable.

    Then the functions G:RR and G:H10(Ω)R given by

    G(ξ)=ξ0g(t)dt,   ξR,G(u)=ΩG(u(x))dx,     uH10(Ω),

    respectively, are well defined and locally Lipschitz continuous. So it makes sense to consider their generalized directional derivatives G0 and G0. On account of [22,formula(9),P.84] one has

    G0(u;v)ΩG0(u(x);v(x))dx, u, vH10(Ω). (5.2)

    We will further assume

    (g2)  limt0g(t)t=0;

    (g3)  lim sup|t|+g(t)t0;

    (g4) there exists a ξ0R such that G(ξ0)<0.

    Through (g3) for every ϵ>0 there exists a constant r>0 such that

    g(t)ϵt, for all |t|r. (5.3)

    Since g is locally bounded, we also have

    M=supt[r,r]|g(t)|<+. (5.4)

    Let λ>0, μ(Ω) be the Lebesgue measure of Ω. Define

    rλ=4λ+2Mrμ(Ω).

    A set KλH10(Ω) is called of type (Kgλ) provided

    (Kgλ):Kλ is convex and closed in H10(Ω). Moreover, ¯BrλKλ.

    Given λ>0 and Kλ satisfying (Kgλ), consider the elliptic variational-hemivariational inequality problems:

    (Pλ): Find uKλ such that for all vKλ,

    Ωu(x)(vu)(x)dxΩa(x)u(x)(vu)(x)dxλG0(u;vu).

    Due to (5.2), any solution u of (Pλ) also fulfills the inequality

    Ωu(x)(vu)(x)dxΩa(x)u(x)(vu)(x)dx
    λΩG0(u(x);(vu)(x))dx, for all  vKλ.

    If g is continuous and Kλ=H10(Ω), the function uH10(Ω) turns out a weak solution to the Dirichlet problem

    {Δu+a(x)u=λg(u),inΩ,u=0,onΩ,

    which has been previously investigated in [3,14] under more restrictive conditions.

    Theorem 5.1 Suppose (g1)(g4) hold true. Then, for every λ sufficiently large, problem (Pλ) possesses at least two solutions.

    Proof. Let X=H10(Ω), p(2,2). Define a functional f(u)=Φ(u)+Ψ(u) on X as follows:

    Φ(u)=12Ω(|u(x)|2+a(x)u(x)2)dx+λG(u)

    as well as

    Ψ(u)={0,  if  uKλ,+,   otherwise,

    where λ>0 and KλH10(Ω) is of type (Kgλ). Owing to (g1) the function Φ:XR turns out locally Lipschitz continuous. Consequently, f satisfies condition (Hf).

    We shall prove that f is bounded from below and coercive.

    By (5.3) and (5.4), one has

    ξ0g(t)dtMr+ϵ2ξ2,  ξR. (5.5)

    Which clearly implies

    G(u)Mrμ(Ω)ϵ2u2L2(Ω),  uX.

    Then we obtain

    f(u)Φ(u)12u2+β2u2L2(Ω)λϵ2u2L2(Ω)λMrμ(Ω)=ϵ2u2+12(βϵλ)u2L2(Ω)λMrμ(Ω).

    Setting ϵ(0,βλ), then we have

    f(u)12u2λMrμ(Ω),   uX, (5.6)

    which shows the claim.

    Let us next show that the function f satisfies condition (PS)f. Pick a sequence {un}X such that {f(un)} is bounded and

    Φ0(un;vun)+Ψ(v)Ψ(un)ϵnvun. (5.7)

    for all  nN,vX, where ϵn0+.

    By (5.7) one evidently has {un}Kλ, and {f(un)} is bounded. Since f is coercive, the sequence {un} turns out bounded. Passing to a subsequence if necessary, we suppose unu in X and unu in L2(Ω). The point u belongs to Kλ because this set is weakly closed.

    Exploiting (5.7) with v=u, we then get

    Ωun(x)(uun)(x)dx+Ωa(x)un(x)(uun)(x)dx+λG0(un;uun)ϵnuun, (5.8)

    for all  nN.

    From unu in X it follows

    limn+Ωa(x)un(x)(uun)(x)dx=0. (5.9)

    The upper semi-continuity of G0 on L2(Ω)×L2(Ω) forces

    lim supn+G0(un;uun)G0(u;0)=0. (5.10)

    Taking account of (5.9), (5.10) besides {uun} is bounded, and letting n+, inequality (5.8) yields

    lim supn+Ω|un(x)|2dxΩ|u(x)|2dx.

    Hence, thanks to [2,Proposition Ⅲ.3], unu in X. i.e., (PS)a holds.

    By (g4), we can construct an u0X such that G(u0)<0. Moreover, u0¯Brλ for any λ14u02. Therefore, infuXf(u)f(u0)<0 provided

    λ>max{14u02,12G(u0)Ω(|u0(x)|2+a(x)u0(x)2)dx},

    while f(0)=λG(0)=0.

    Our next objective is to verify (i1). From (g2), there exists σ(0,r) such that

    |u(x)|<σ[u(x)0g(t)dt]dxϵ2Ω|u(x)|2dx. (5.11)

    Due to (5.5), one has

    G(ξ)Mrϵ2ξ2(Mrσp+ϵ2σp2)|ξ|p,

    provided |ξ|σ.

    The Sobolev embedding theorem gives

    |u(x)|σG(u(x))dx(Mrσp+ϵ2σp2)upLp(Ω)Cup, (5.12)

    where C=(Mrσp+ϵ2σp2)Cpp. Then by (5.11) (5.12) and (5.1) we get

    G(u)=|u(x)|<σ[u(x)0g(t)dt]dx+|u(x)|σG(u(x))dxϵ2C22u2Cup=u2(ϵ2C22+Cup2),uX. (5.13)

    Let us next prove that for a suitable constant θ>0,

    Ω(|u(x)|2+a(x)u(x)2)dxθΩ|u(x)|2dx,  uX. (5.14)

    Indeed, if it's not true, there exists a sequence {un}X enjoying the properties

    un=1, nN,
    Ω(|un(x)|2+a(x)un(x)2)dx<1n,  nN. (5.15)

    Passing to a subsequence if necessary, we may suppose unu in X as well as unu in L2(Ω). Thus, letting n+ in (5.15) yields

    Ω(|u(x)|2+a(x)u(x)2)dx0. (5.16)

    Using the sobolev embedding theorem and β=essinfxΩa(x)0 we obtain

    (1C22+β)u2L2(Ω)Ω(|u(x)|2+a(x)u(x)2)dx. (5.17)

    Gathering (5.16) and (5.17) together, leads to u=0. By (5.15) this forces un0 in X, against to un=1,nN.

    Combining (5.14) with (5.13), provides

    f(u)u2(θ2λ(ϵ2C22+Cup2)),  uX. (5.18)

    Pick ϵ>0 and R(0,12u0) sufficiently small such that

    θ2λ(ϵ2C22+CRp2)>0.

    Then by (5.18) we have

    f(u)0,  u¯BR. (5.19)

    Furthermore, it is easy to prove that R<12u0<rλ.

    Now, let v0=0, v1=u0. Define

    Γ={γC0([0,1],X):γ(0)=v0, γ(1)=v1},c=infγΓsupt[0,1]f(γ(t)).

    Thanks to (5.19) and the definition of c, one has

    c0=max{f(v0), f(v1)},

    and for every γΓ, there exists a t(0,1) such that γ(t)X and γ(t)=R. Then by (5.19) again, we obtain f(γ(t))0. Hence hypothesis (i1) of Theorem 4.1 is fulfilled.

    Finally, let us prove that (i3) holds. Since f is bounded below, put α=infxXf(x), then α<0c. For every aα suppose that a<λ, then there exist r>0 and ϵ0>0 such that

    Fra,ϵ0int(DΨ). (5.20)

    Indeed, there is ϵ0>0 such that a+ϵ0λ<2λ.

    Inequality (5.6) ensures that

    {uX:f(u)a+ϵ0}{uX:f(u)λ}{uX:u<rλ}¯BrλDΨ.

    So we immediately have {uX:f(u)a+ϵ0}int(DΨ).

    Since f is coercive, there exists r>0 such that every uX satisfies aϵ0f(u)a+ϵ0, and ur+1, which leads to (5.20), i.e., condition (i3) holds true.

    We are now in a position to apply Theorem 4.1. By this theorem, there exist at least two points u1,u2X such that

    Φ0(ui;vui)+Ψ(v)Ψ(ui)0,  vX, i=1,2.

    The choice of Ψ gives both uiKλ and Φ0(ui;vui)0, vKλ, i=1,2. Namely, u1,u2 are solutions to the problem (Pλ).

    Example 5.2 The aim of this example is to exhibit a nontrivial case of set in H10(Ω) of type (Kgλ). Let h:H10(Ω)R be a weakly continuous and convex function. For ˉr>0 fixed, λ>0, put

    ˉrλ=4λ+2Mˉrμ(Ω),

    with the same notation as before. The ball ˉB(0,ˉrλ) is a weakly compact subset of H10(Ω), since h is weakly continuous, there exists u0ˉB(0,ˉrλ) such that

    γ=maxuˉB(0,ˉrλ)h(u)=h(u0),

    i.e., hˉB(0,ˉrλ) admits a global maximum. Then the set

    Kλ:={uH10(Ω) : h(u)γ+1}

    is a subset of H10(Ω) of type (Kgλ).

    Example 5.3 There exist functionals satisfying the conditions of Theorem 5.1. For example

    g(t)={|t|(1et2),|t|1,t(et21),  |t|>1.

    The authors are grateful to the anonymous referees for their careful reading and helpful comments, which greatly improve the manuscript. The work was supported by the NSF of Shandong Province (No. ZR2018PA006, ZR2017PA001) and the NSF of China (No. 11901240).

    The authors declare no conflict of interest.



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